Would this be fairly good for far distances?
Depends on what you mean by "far". I really recommend that you borrow or somehow check out a few lenses of different lengths before you buy. I had the 55-250, which is a VERY good deal, and I now use the 70-200 f/4 L IS (a fabulous lens, but not near your budget) with a 1.4x teleconverter when I need it, for a total of 280mm. Even on a crop sensor sensor camera (everything up to a 7D), which gives you more reach than a full frame, 280mm is not long enough for a some wildlife uses. E.g., I tried a bunch of loon shots a few weeks ago with that setup, and I could not get close enough. for larger animals you might make do, and for things that let you get closer. You need to see for yourself whether that focal length range will do for you.
If it will, the 55-250 is a great deal as a starter lens. It is very cheap, amazingly light, and has pretty good IQ.